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Mirrors > Home > MPE Home > Th. List > abl32 | Structured version Visualization version GIF version |
Description: Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
ablcom.b | ⊢ 𝐵 = (Base‘𝐺) |
ablcom.p | ⊢ + = (+g‘𝐺) |
abl32.g | ⊢ (𝜑 → 𝐺 ∈ Abel) |
abl32.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
abl32.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
abl32.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
Ref | Expression |
---|---|
abl32 | ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abl32.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
2 | ablcmn 18199 | . . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
4 | abl32.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | abl32.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | abl32.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
7 | ablcom.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
8 | ablcom.p | . . 3 ⊢ + = (+g‘𝐺) | |
9 | 7, 8 | cmn32 18211 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) |
10 | 3, 4, 5, 6, 9 | syl13anc 1328 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 CMndccmn 18193 Abelcabl 18194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-sgrp 17284 df-mnd 17295 df-cmn 18195 df-abl 18196 |
This theorem is referenced by: matunitlindflem1 33405 baerlem5alem1 36997 baerlem5blem1 36998 |
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